2004 Fiscal Year Final Research Report Summary
Degeneration of closed Riemann surfaces and limits of Jacobi varieties
| Project/Area Number |
14540163
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Toyama University |
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Principal Investigator |
ABE Yukitaka Toyama University, Faculty of Science, Math, Professor, 理学部, 教授 (80167949)
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| Co-Investigator(Kenkyū-buntansha) |
AZUKAWA Kazuo Toyama University, Faculty of Science, Math, Professor, 理学部, 教授 (20018998)
KODA Takashi Toyama University, Faculty of Science, Math, Associate Professor, 理学部, 助教授 (40215273)
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| Project Period (FY) |
2002 – 2004
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| Keywords | closed Riemann surface / jacobi variety / theta constant / Satake compactification / moduli space / degenerate abelian function |
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| Research Abstract |
Any closed Riemann surface of genus 2 is regarded as a double covering surface over the one-dimensional complex projective space with 6 branch points. We fix one of the branch points at the infinity. We found out a way to represent ratios of differences of these branch points by theta constants. We investigated behaviour of the ratios when Jacobi varieties of closed Riemann surfaces tend to limits. Then we recognized which curves are regarded as limits of degeneration.
We also considered limits of Jacobi varieties. Compactification of the moduli space of abelian varieties was first given by Satake in 1956. It is very natural to require that limits of abelian varieties have function fields as limits of abelian function fields. These function fields are just what Weierstrass stated. We could determine them. From this point of view, the Satake compactification is the most natural one. Then we considered limits of Jacobi varieties in the Satake compactification in the above.
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